(b) after 31 October 1998, whenever a lease is commenced, or terminated and renewed, whether with an existing or new tenant, reg 36(3)(b) requires landlords to ensure that a safety check has been carried out on any 'relevant appliance' within the 12 month period before any new lease commences, or within 12 months of any new appliance and associated flue having been installed in the premises to be leased, whichever is the later.
Could be wrong but does Reg 36 say that a landlord should undertake a LLSC on a new lease?
No. Read the bold/italic bits carefully.
This means that, where the new tenancy continues straight on from the old one, even if it is a new tenant, the existing certificate is valid, provided it is not more than 12 months old. But, if there is a gap between tenants and the certificate has expired, then a new certificate will be required when the new tenancy commences.